Question

which equation can be solved using the expression \\(\frac{-3\pm\sqrt{(3)^2 + 4(10)(2)}}{2(10)}\\) for x?\
\\(\bigcirc\\) \\(10x^2 = 3x + 2\\)\
\\(\bigcirc\\) \\(2 = 3x + 10x^2\\)\
\\(\bigcirc\\) \\(3x = 10x^2 - 2\\)\
\\(\bigcirc\\) \\(10x^2 + 2 = -3x\\)

Explanation:

Step1: Recall Quadratic Formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). But in the given expression, we have \(+ 4ac\) inside the square root, which means the equation is of the form \(ax^{2}+bx + c = 0\) rearranged such that we have \(ax^{2}+bx=-c\) or other forms, but when we bring all terms to left, if \(c\) was negative, then \(- 4ac\) becomes \(+4|c|a\). Wait, actually, let's compare the given expression \(\frac{-3\pm\sqrt{(3)^{2}+4(10)(2)}}{2(10)}\) with the quadratic formula. Let's rewrite the quadratic formula as \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). If we have \(+4ac\) instead of \(-4ac\), that means \(c\) is negative in the standard form. So let's assume the equation is \(ax^{2}+bx + c = 0\), then the discriminant is \(b^{2}-4ac\), but in our case, it's \(b^{2}+4ac\), so that implies \(c\) is negative, i.e., the equation is \(ax^{2}+bx - |c|=0\), so discriminant is \(b^{2}+4a|c|\). Now, from the given expression, the numerator is \(-3\pm\sqrt{(3)^{2}+4(10)(2)}\) and denominator is \(2(10)\). So comparing with quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), we can see that \(a = 10\), \(b = 3\)? Wait no, wait the numerator is \(-3\pm\sqrt{...}\), so \(-b=-3\) implies \(b = 3\)? Wait no, wait the standard formula is \(\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). So if our numerator is \(-3\pm\sqrt{(3)^{2}+4(10)(2)}\), that means \(-b=-3\) so \(b = 3\), and \(2a = 2(10)\) so \(a = 10\), and \(-4ac=+4(10)(2)\) which implies \(-4ac = 80\), so \(c=-2\) (since \(a = 10\), then \(-4*10*c=80\) => \(-40c = 80\) => \(c=-2\)). So the quadratic equation should be \(ax^{2}+bx + c = 0\) => \(10x^{2}+3x-2 = 0\)? Wait no, wait let's check each option by rearranging them into standard form \(ax^{2}+bx + c = 0\) and see which one gives \(a = 10\), \(b = 3\), \(c=-2\) or similar.

Step2: Analyze each option

- Option 1: \(10x^{2}=3x + 2\) => \(10x^{2}-3x - 2=0\). Here \(a = 10\), \(b=-3\), \(c=-2\). Then quadratic formula would be \(\frac{-(-3)\pm\sqrt{(-3)^{2}-4(10)(-2)}}{2(10)}=\frac{3\pm\sqrt{9 + 80}}{20}\), which is not the given expression.
- Option 2: \(2 = 3x+10x^{2}\) => \(10x^{2}+3x - 2=0\). Here \(a = 10\), \(b = 3\), \(c=-2\). Then discriminant is \(b^{2}-4ac=(3)^{2}-4(10)(-2)=9 + 80\), and quadratic formula is \(\frac{-3\pm\sqrt{9 + 80}}{2(10)}\), which matches the given expression \(\frac{-3\pm\sqrt{(3)^{2}+4(10)(2)}}{2(10)}\) (since \(-4ac=-4*10*(-2)=80 = 4*10*2\)). Wait, but let's check other options.
- Option 3: \(3x=10x^{2}-2\) => \(10x^{2}-3x - 2=0\). Here \(a = 10\), \(b=-3\), \(c=-2\). Quadratic formula: \(\frac{-(-3)\pm\sqrt{(-3)^{2}-4(10)(-2)}}{2(10)}=\frac{3\pm\sqrt{9 + 80}}{20}\), not matching.
- Option 4: \(10x^{2}+2=-3x\) => \(10x^{2}+3x + 2=0\). Here \(a = 10\), \(b = 3\), \(c = 2\). Then discriminant is \(b^{2}-4ac=9-80=-71\), and the expression would be \(\frac{-3\pm\sqrt{9 - 80}}{2(10)}\), which is not the given one. Wait, but wait in option 2, the equation is \(10x^{2}+3x - 2=0\), so \(c=-2\), so \(-4ac=-4*10*(-2)=80=4*10*2\), so the discriminant is \(b^{2}+4*10*2\) (since \(-4ac = +4*10*2\) when \(c=-2\)). So the quadratic formula for \(10x^{2}+3x - 2=0\) is \(x=\frac{-3\pm\sqrt{3^{2}-4*10*(-2)}}{2*10}=\frac{-3\pm\sqrt{9 + 80}}{20}=\frac{-3\pm\sqrt{3^{2}+4*10*2}}{2*10}\), which matches the given expression. Wait, but let's check the option 2 again. The option 2 is \(2 = 3x + 10x^{2}\), which rearranged is \(10x^{2}+3x - 2=0\), correct. Wait but also, let's check option 4: \(10x^{2}+2=-3x\) => \(10x^{2}+3x + 2=0\), which has \(c = 2\), so discriminant is \(9-80\), not matching. Option 1: \(10x^{2}-3x - 2=0\), \(b=-3\), so numerator is \(3\pm...\), not \(-3\pm...\). Option 3: \(10x^{2}-3x - 2=0\), same as option 1. Option 2: \(10x^{2}+3x - 2=0\), so \(b = 3\), so \(-b=-3\), numerator is \(-3\pm...\), denominator \(2*10\), and discriminant \(3^{2}-4*10*(-2)=9 + 80=3^{2}+4*10*2\), which matches the given expression. Wait, but also, let's check option 4 again. Wait the given expression has \(+4(10)(2)\) in the square root, which is 80, and in option 2, the discriminant is \(9 + 80\), which is correct. In option 4, the discriminant would be \(9-80\), which is negative, and the expression has a plus, so option 2 is correct? Wait no, wait the user's options: let's re-express each option in standard form \(ax^{2}+bx + c = 0\):

1. \(10x^{2}-3x - 2=0\) (a=10, b=-3, c=-2)
2. \(10x^{2}+3x - 2=0\) (a=10, b=3, c=-2)
3. \(10x^{2}-3x - 2=0\) (same as 1)
4. \(10x^{2}+3x + 2=0\) (a=10, b=3, c=2)

Now, the given expression is \(\frac{-3\pm\sqrt{3^{2}+4(10)(2)}}{2(10)}\). Let's compute the discriminant part: \(3^{2}+4(10)(2)=9 + 80=89\). Now, for the quadratic formula, discriminant is \(b^{2}-4ac\). So for option 2: \(b^{2}-4ac=3^{2}-4(10)(-2)=9 + 80=89\), which matches the discriminant in the given expression (since \(3^{2}+4(10)(2)=9 + 80=89\)). So the quadratic formula for option 2 is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-3\pm\sqrt{9 + 80}}{20}=\frac{-3\pm\sqrt{3^{2}+4(10)(2)}}{2(10)}\), which is exactly the given expression. Wait, but also, let's check option 4: \(10x^{2}+3x + 2=0\), discriminant is \(3^{2}-4(10)(2)=9 - 80=-71\), which is not equal to \(3^{2}+4(10)(2)\). So option 2 is correct? Wait no, wait the user's option 4 is \(10x^{2}+2=-3x\), which is \(10x^{2}+3x + 2=0\), c=2. Option 2 is \(2 = 3x + 10x^{2}\) => \(10x^{2}+3x - 2=0\), c=-2. So in the quadratic formula, for option 2, \(c=-2\), so \(-4ac=-4*10*(-2)=80=4*10*2\), so the discriminant is \(b^{2}-4ac=b^{2}+4*10*2\) (since c is negative). So that's why the square root has \(+4(10)(2)\). So the equation that can be solved with that expression is option 2? Wait no, wait let's check the numerator: \(-3\pm\sqrt{...}\), so \(b = 3\), because \(-b=-3\) => \(b=3\). So in option 2, \(b=3\), correct. In option 4, \(b=3\) as well, but \(c=2\), so \(-4ac=-80\), so discriminant is \(9-80\), which is not \(9 + 80\). So option 2 is correct. Wait, but let's re-express the given expression. The given expression is \(\frac{-3\pm\sqrt{(3)^{2}+4(10)(2)}}{2(10)}\). Let's write the quadratic formula for each option:

- Option 1: \(10x^{2}-3x - 2=0\). Quadratic formula: \(\frac{3\pm\sqrt{9 + 80}}{20}\) (since \(b=-3\), so \(-b=3\)). Doesn't match.
- Option 2: \(10x^{2}+3x - 2=0\). Quadratic formula: \(\frac{-3\pm\sqrt{9 + 80}}{20}\) (since \(b=3\), so \(-b=-3\)). Matches.
- Option 3: \(10x^{2}-3x - 2=0\). Same as option 1. Doesn't match.
- Option 4: \(10x^{2}+3x + 2=0\). Quadratic formula: \(\frac{-3\pm\sqrt{9 - 80}}{20}\). Doesn't match (has minus, not plus, and discriminant is negative).

So the correct option is the second one, which is \(2 = 3x + 10x^{2}\), i.e., option B (assuming the options are labeled A, B, C, D with A being first, B second, etc. Wait the user's options: first option is \(10x^{2}=3x + 2\) (A), second is \(2 = 3x + 10x^{2}\) (B), third is \(3x = 10x^{2}-2\) (C), fourth is \(10x^{2}+2 = -3x\) (D). So the correct answer is B. \(2 = 3x + 10x^{2}\)

Answer:

B. \(2 = 3x + 10x^{2}\)